The infinite first quadrant of the plane (left) is mapped conformally onto a bounded triangle (right). The two edges of the quadrant become the two legs of the triangle, and the point at infinity is stretched out into the whole hypotenuse. Angles are preserved. Drag in either region to draw a curve and see its image.
A modern JavaScript + Canvas port of a 1998 Java applet by Terence Tao
(originally Compact.java).
Drag near the top-right of the plane to approach the hypotenuse (infinity).